sparse-ir 0.96.0

intermediate representation (IR) basis for electronic propagator

This library provides routines for constructing and working with the intermediate representation of correlation functions. It provides:

• on-the-fly computation of basis functions for arbitrary cutoff Λ

• basis functions and singular values are accurate to full precision

• routines for sparse sampling

Installation

Install via pip:

pip install sparse-ir[xprec]

The above line is the recommended way to install sparse-ir. It automatically installs the xprec package, which allows to compute the IR basis functions with greater accuracy. If you do not want to do this, simply remove the string [xprec] from the above command.

Install via conda:

conda install -c spm-lab sparse-ir xprec

Quick start

Check out our comprehensive tutorial!

Here is some python code illustrating the API:

# Compute IR basis for fermions and β = 10, W <= 4.2
import sparse_ir, numpy
basis = sparse_ir.FiniteTempBasis(statistics='F', beta=10, wmax=4.2)

# Assume spectrum is a single pole at ω = 2.5, compute G(iw)
# on the first few Matsubara frequencies. (Fermionic/bosonic Matsubara
# frequencies are denoted by odd/even integers.)
gl = basis.s * basis.v(2.5)
giw = gl @ basis.uhat([1, 3, 5, 7])

# Reconstruct same coefficients from sparse sampling on the Matsubara axis:
smpl_iw = sparse_ir.MatsubaraSampling(basis)
giw = -1/(1j * numpy.pi/basis.beta * smpl_iw.wn - 2.5)
gl_rec = smpl_iw.fit(giw)

You may want to start with reading up on the intermediate representation. It is tied to the analytic continuation of bosonic/fermionic spectral functions from (real) frequencies to imaginary time, a transformation mediated by a kernel K. The kernel depends on a cutoff, which you should choose to be lambda_ >= beta * W, where beta is the inverse temperature and W is the bandwidth.

One can now perform a singular value expansion on this kernel, which generates two sets of orthonormal basis functions, one set v[l](w) for real frequency side w, and one set u[l](tau) for the same obejct in imaginary (Euclidean) time tau, together with a “coupling” strength s[l] between the two sides.

By this construction, the imaginary time basis can be shown to be optimal in terms of compactness.

Refer to the online documentation for more details.

License and citation

This software is released under the MIT License. See LICENSE.txt for details.

If you find the intermediate representation, sparse sampling, or this software useful in your research, please consider citing the following papers:

• Hiroshi Shinaoka et al., Phys. Rev. B 96, 035147 (2017)

• Jia Li et al., Phys. Rev. B 101, 035144 (2020)

• Markus Wallerberger et al., arXiv 2206.11762 (2022)

If you are discussing sparse sampling in your research specifically, please also consider citing an independently discovered, closely related approach, the MINIMAX isometry method (Merzuk Kaltak and Georg Kresse, Phys. Rev. B 101, 205145, 2020).